Stretched exponentials withT-dependent exponents from fixed distributions of energy barriers for relaxation times in fast-ion conductors

Abstract

The conductivity modulus in fast ion conductors (FIC) has often been fitted with stretched exponential relaxation functions of the form $\exp \left[-\right(t/{\tau }_K{)}^{\beta }],$ where the β values are taken to be independent of temperature. This analysis corresponds to the assumption of an asymmetric distribution of relaxation times (DRT) that does not have the T dependence observed in the conductivity modulus spectra found for many FIC glasses and expected from a fixed distribution of activation energies (DAE) against the ion hops. In this paper, it is shown, rather, that a fixed DAE leads to the temperature dependence of $\beta \propto T^{0.5}$ for β values near 0.6, and conductivity modulus data for some ${\mathrm{Li}}^{+}$ glasses are presented where β varies nearly in this way. Similar behavior can be inferred from published data for many other FIC. It is also shown that the DRT calculated from the corresponding stretched exponential relaxation function closely approximates the tail of the DRT calculated from the DAE and truncated at the conductivity percolation limit of 0.3.